Question

4.1 The Butcher-Gauss Runge-Kutta method The aim of this project is to compare the implicit Butcher-Gauss Runge-Kutta method with the classical explcit Runge-Kutta method, mainly through numerical simulations. Consider the 2-stage implicit Runge-Kutta method given by the following Butcher tableau: (C.1) \begin{array}{c|cc} 1/2 - \sqrt{3}/6 & 1/4 & 1/4 - \sqrt{3}/6 \\ 1/2 + \sqrt{3}/6 & 1/4 + \sqrt{3}/6 & 1/4 \\ \hline & 1/2 & 1/2 \end{array} This method is known as the 4-th order Gauss-Butcher Runge-Kutta (GBRK4) method. (a) Show that GBRK4 is indeed a 4-th order method, thus justifying its name.

          4.1 The Butcher-Gauss Runge-Kutta method
The aim of this project is to compare the implicit Butcher-Gauss Runge-Kutta method with
the classical explcit Runge-Kutta method, mainly through numerical simulations.
Consider the 2-stage implicit Runge-Kutta method given by the following Butcher tableau:
(C.1)
\begin{array}{c|cc}
1/2 - \sqrt{3}/6 & 1/4 & 1/4 - \sqrt{3}/6 \\
1/2 + \sqrt{3}/6 & 1/4 + \sqrt{3}/6 & 1/4 \\
\hline
 & 1/2 & 1/2
\end{array}
This method is known as the 4-th order Gauss-Butcher Runge-Kutta (GBRK4) method.
(a) Show that GBRK4 is indeed a 4-th order method, thus justifying its name.

4.1 The Butcher-Gauss Runge-Kutta method
The aim of this project is to compare the implicit Butcher-Gauss Runge-Kutta method with
the classical explcit Runge-Kutta method, mainly through numerical simulations.
Consider the 2-stage implicit Runge-Kutta method given by the following Butcher tableau:
(C.1)

1/2 - √(3)/6 1/4 1/4 - √(3)/6

1/2 + √(3)/6 1/4 + √(3)/6 1/4
1/2 1/2

This method is known as the 4-th order Gauss-Butcher Runge-Kutta (GBRK4) method.
(a) Show that GBRK4 is indeed a 4-th order method, thus justifying its name.

Added by Jose Maria G.

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